A Treatise on Analytic Geometry; Especially as Applied to the Properties of Conics Including the Modern Methods of Abridged Notation

A Treatise on Analytic Geometry; Especially as Applied to the Properties of Conics Including the Modern Methods of Abridged Notation

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1869 edition. Excerpt: ...a distance we may assign as the true one, the points will have drawn nearer together in the very instant in which we assign it: so that their distance eludes all attempts at finite statement, and can only be represented by the phrase infinitely small. The geometric meaning of this analytic conception varies with its different applications. Thus, it may signify exactly the same thing as the single point which, in the language of pure geometry, is common to a curve and its tangent. For since the distance between consecutive points is infinitely small; that is, so small that we can not assign a value too small for it; we may assign the value 0, and take the points as absolutely coincident. It is in this aspect, mainly, that we shall use the conception in our future inquiries. Hence, as from the infinite series of continuous values which the distance between two consecutive points must have, we thus select the one corresponding to the moment of coincidence, we have preferred to designate the conception by the equivalent and for us more pertinent phrase coincident points. The student should be sure that he always thinks of the distance between coincident points as a true infinitesimal. The error into which the beginner almost always falls is, to think of a very small, instead of an infinitely small, distance. He thus confounds with two consecutive points, two discrete ones extremely close together, between which there is of course a finite distance. The consequence is, that he finds in such a distance, however small, an infinite number of points lying between his supposed consecutives, and fancies that all the arguments based on the conception of consecutiveness are fallacious. Whereas, if he excludes from his thoughts, as he should, all points...show more

Product details

  • Paperback | 138 pages
  • 189 x 246 x 8mm | 259g
  • Rarebooksclub.com
  • Miami Fl, United States
  • English
  • black & white illustrations
  • 1236634012
  • 9781236634016